H-distribution via Sobolev spaces

TL;DR

This paper characterizes H-distributions associated with weakly convergent sequences in Sobolev spaces and shows their relation to strong convergence, with applications to solutions of linear PDEs.

Contribution

It provides a complete characterization of H-distributions for weakly convergent Sobolev space sequences and links their vanishing to strong convergence properties.

Findings

H-distributions are zero iff the sequence converges strongly after localization.

The results apply to weakly convergent solutions of linear first order PDEs.

Characterization aids in understanding convergence behavior in Sobolev spaces.

Abstract

H-distributions associated to weakly convergent sequences in Sobolev spaces are determined. It is shown that a weakly convergent sequence $(u_n)$ in $W^{-k,p}( \R^d)$ has the property that $\theta u_n$ converges strongly in $W^{-k,p}(\R^d)$ for every $\theta\in\mathcal S(\R^d)$ if and only if all H-distributions related to this sequence are equal to zero. Results are applied on a weakly convergent sequence of solutions to a family of linear first order PDEs.